# Improving Fraction Teaching with a Game: FRAX!

This post is the second on the game FRAX and spends some time explaining how the game works and where it came from. The first post was on Occupy Math’s sister blog Dan and Andrew’s Game Place. FRAX seems, based on our initial testing, to be a fun game in addition to giving the players practice with fraction arithmetic. To get the rules to FRAX (and if you’re interested in testing the game), click the link! We are giving away FRAX sets to people who will help us with the play testing. FRAX is a card game, not a computer game, though we have some thoughts in that direction.

Why do we Need FRAX?

The short answer is because the fraction teaching we are doing now is not working. An earlier Occupy Math, How Not to Teach Fractions gives some perspective on the problems we are currently facing, but the Jack Acid test that there is a problem is that many of the students who come into Occupy Math’s first year university class cannot do arithmetic on fractions reliably — and this is as we hand them problems like fractions jammed with functions and variables that show up naturally in first-year calculus.

Occupy Math has a long experience with enhancing math teaching with competition and has been a hobby-level game designer for decades. FRAX, designed in collaboration with Andrew McEachern, a teaching-focused professor at the University of Victoria, leverages this experience to create a fraction teaching game that is legitimately a fun game in addition to teaching fractions. I’d also like to shout out to Bradley Rinke who is responsible for the striking art work on the cards, like the card-back at the top of the post.

Caveat: FRAX will help students get better at fractions. It cannot do the whole job of teaching! FRAX is a family of card games where, while you don’t have to know fractions to play, being able to work with fractions gives you an advantage in the game. This creates a soft on-ramp to improving your fraction skills.

The History of FRAX

In 2011, Elizabeth Knowles, a student who had taken some of Occupy Math’s courses, came by looking for a topic for a senior project. After a long conversation we decided to investigate what would happen if we took Prisoner’s Dilemma and turned it into a card game by printing the two moves, cooperate and defect, on cards. Prisoner’s dilemma models the situation where two prisoners, that were partners in a crime, are put in different rooms and given a chance to testify against one another. Their possible moves are to cooperate (maintain silence) or defect (testify against their partner).

The obvious difference is that, instead of being able to make any move you like, the cards you have in your hand would limit your choices. How much does it matter that you can’t always make any move you want? It turns out that one change creates a whole lot of other changes.

One way you can tell a project is a good one is if you manage to publish a paper about the results. The link leads to a joint paper with Ms. Knowles on Deck Based Prisoner’s Dilemma. What we found out is this:

• The game splits into three different games, none of which are Prisoner’s Dilemma, depending on the exact numerical values used to score the game.
• One of the games has the optimal strategy “play the same move as your opponent”.
• Another has the optimal strategy “play the opposite move from your opponent”.
• The last game — which is in a very small part of the space of possible payoffs — has the remarkably odd quality that your final score is completely independent of how you play.
• When one of the two moves, cooperate or defect, is substantially more costly than the other for a player to make, the deck-based Prisoner’s Dilemma is a simple model of situations that actually arise in practice, like contributing funds toward a commodity promotion board.

Given that Prisoner’s Dilemma, which is about as simple as a game can get, suddenly gains all sorts of nuance and complexity when you make it a card game, it is natural to ask what happens to other mathematical games. Justin Shonfeld and Occupy Math published a paper on the general theory of turning mathematical games into card games in the new journal Game and Puzzle Design. This article talks about card game versions of Rock-Paper-Scissors and other whimsical topics. The real payoff came when we looked at a game that looked too hard to turn into a card game.

Divide-the-Dollar and FRAX

Occupy Math has been working with the game Divide the Dollar, invented by John Nash, for over twenty years. In this game, two players submit bids. If the bids total a dollar or less, both players receive their bids; otherwise, they get nothing. The game is a simple model of making a deal. It also has 101 possible moves for each player, making it really tricky to work with using the techniques used on Prisoner’s Dilemma or Rock-Paper-Scissors. For example, if you want three cards for each possible move, you need 303 cards. Ouch!

Solving this problem is the key to FRAX. Instead of allowing free, open play with all fractions of a dollar down to the penny, use only relatively simple fractions. Limiting which numbers may be used in play yields a playable card game (well, several playable card games, but you get the idea). There are still a bunch of things we had to do to create FRAX, but deck-based Divide-the-Dollar is the core notion.

The FRAX game expanded Divide-the-Dollar to do the following:

• Allow more than two players. For this we need to be able to bid for more than one dollar — and the goal cards do exactly this.
• Allow turn-based or simultaneous play (there are two sets of rules).
• Added a social component that teaches the skills that lead up to doing mathematical proofs — or speaking effectively to an audience.
• Permit working with fractions in multiple ways or representations. These include classic fractions, dot clouds, and pizza slices. Andrew McEachern has written an explanation of these.

Once you hit on the basic idea of FRAX you still have some other issues. These include the problem of selecting the set of cards to use and checking that your game design works properly. That is why we need beta testers.

This post is, frankly, part of a commercial venture. Occupy Math is trying to raise funds by producing useful solutions to the problems he writes about. This is part of the philosophy of trying to not just complain or express outrage, but to also propose solutions. Occupy Math and Dr. McEachern will be selling FRAX as a game once we finish beta testing and any updates it suggests. Do you have suggestions for projects you would like to see Occupy Math develop? Please comment or tweet!

I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics